M2 June 2014 (R) Q2
2. A ball of mass 0.4 kg is moving in a horizontal plane when it is struck by a bat. The bat exerts an impulse \((-5\mathbf{i} + 3\mathbf{j})\) N s on the ball. Immediately after receiving the impulse the ball has velocity \((12\mathbf{i} + 15\mathbf{j})\) m s\(^{-1}\).
Find
| Scheme | Marks |
|---|---|
| \(0.4\mathbf{u} + (-5\mathbf{i} + 3\mathbf{j}) = 0.4(12\mathbf{i} + 15\mathbf{j})\) | M1 A1 |
| \(\mathbf{u} = \dfrac{9.8\mathbf{i} + 3\mathbf{j}}{0.4} = 24.5\mathbf{i} + 7.5\mathbf{j}\) | |
| Speed \(= \sqrt{24.5^2 + 7.5^2} = 25.6\) m s\(^{-1}\) | M1 A1 |
| (4) |
Notes
M1 Impulse-momentum equation. Requires all 3 terms. Must be dimensionally correct. Condone sign error(s).
A1 Correct unsimplified equation
M1 Use of Pythagoras’ theorem to find magnitude of their \(\mathbf{u}\)
A1 25.6 or better
| Scheme | Marks |
|---|---|
| \(\tan^{-1}\left(\dfrac{15}{12}\right),\ \ \tan^{-1}\left(\dfrac{7.5}{24.5}\right)\) | M1 A1 |
| \(= 34.319\ldots^\circ\) or \(0.59899\ldots\) rad | A1 |
| (3) | |
| (7 marks) |
Notes
M1 Use trig. to find useful angles. Follow their \(\mathbf{u}\)
A1 Correct unsimplified. (51.3\(^\circ\) & 17.0\(^\circ\), 38.7\(^\circ\) & 73\(^\circ\))
A1 Combine correctly to find the required angle. 34\(^\circ\), 0.60 rads or better
Alt(b)
| \(\cos\theta = \dfrac{\begin{pmatrix}24.5\\7.5\end{pmatrix}\cdot\begin{pmatrix}12\\15\end{pmatrix}}{\sqrt{24.5^2 + 7.5^2}\sqrt{12^2 + 15^2}}\) | M1 A1 |
| \(\theta = 34.319\ldots^\circ\) or \(0.59899\ldots\) rad | A1 |
M1 Use scalar product. Follow their \(\mathbf{u}\)
A1 Correctly substituted.